document.write( "Question 189647: determine whether the function f(x)=3x^3-5x is even, odd, or neither. b) name the graph family for this function. \n" ); document.write( "

Algebra.Com's Answer #142300 by jim_thompson5910(35256) $\"\"$ $\"About$

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a)\r
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\n" ); document.write( "\n" ); document.write( "Remember, if $\"f%28x%29=f%28-x%29\"$ then the function is an even function. If $\"f%28-x%29=-f%28x%29\"$ then the function is an odd function.\r
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\n" ); document.write( "\n" ); document.write( "First, let's see if $\"f%28x%29=3x%5E3-5x\"$ is an even function.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=3x%5E3-5x\"$ Start with the given function.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-x%29=3%28-x%29%5E3-5%28-x%29\"$ Replace each x with -x.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-x%29=-3x%5E3%2B5x\"$ Simplify. Note: only the terms with an odd exponent will change in sign.\r
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\n" ); document.write( "\n" ); document.write( "So this shows us that $\"3x%5E3-5x%3C%3E-3x%5E3%2B5x\"$ which means that $\"f%28x%29%3C%3Ef%28-x%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since $\"f%28x%29%3C%3Ef%28-x%29\"$ , this shows us that $\"f%28x%29=3x%5E3-5x\"$ is not an even function.\r
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\n" ); document.write( "\n" ); document.write( "Now, let's see if $\"f%28x%29=3x%5E3-5x\"$ is an odd function.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=3x%5E3-5x\"$ Start with the given function.\r
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\n" ); document.write( "\n" ); document.write( " $\"-f%28x%29=-%283x%5E3-5x%29\"$ Negate the entire function by placing a negative outside the function.\r
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\n" ); document.write( "\n" ); document.write( " $\"-f%28x%29=-3x%5E3%2B5x\"$ Distribute and simplify.\r
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\n" ); document.write( "\n" ); document.write( "So this shows us that $\"-3x%5E3%2B5x=-3x%5E3%2B5x\"$ which means that $\"f%28-x%29=-f%28x%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since $\"f%28-x%29=-f%28x%29\"$ , this shows us that $\"f%28x%29=3x%5E3-5x\"$ is an odd function.\r
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\n" ); document.write( "\n" ); document.write( "Note: if you graph $\"f%28x%29=3x%5E3-5x\"$ , you'll find that the graph has symmetry about the origin.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So the function $\"f%28x%29=3x%5E3-5x\"$ is an odd function.\r
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\n" ); document.write( "\n" ); document.write( "b) Since the degree of $\"f%28x%29=3x%5E3-5x\"$ is 3, this means that this graph is part of a family of cubics. \n" ); document.write( "

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